Fig. 4: Latent ideal observer accounts for dissociations between decisions and confidence.

a Denoising variational autoencoder (VAE) used to extract low-dimensional latent representation of training data. An image x was passed through a DNN encoder q (distinct from the encoder f used in the supervised neural network model), which output the parameters (means and variances) of the latent posterior q(z∣x), a two-dimensional Gaussian distribution. This distribution was sampled from, yielding z, which was then passed through a DNN decoder h, yielding \(\tilde{{{{{{{{\bf{x}}}}}}}}}\), a denoised reconstruction of the input x. The VAE was regularized based on the divergence of the latent posterior from a unit normal prior (with means equal to 0 and variances equal to 1), encouraging efficient low-dimensional encodings of the high-dimensional inputs. b Latent ideal observer model. After training the VAE, Gaussian distributions were fit to the latent representations resulting from the training images for classes s1 and s2. The distributions were used to construct an ideal observer model that computed confidence according to p(correct∣z), the probability of being correct given the low-dimensional embedding z. Concentric ellipses represent distributions based on the average parameters of those extracted from 100 trained VAEs. The latent ideal observer accounted for both versions of the PE bias, including (c) the version involving manipulation of contrast and noise (Fig. 3a; two-sided paired t-tests, accuracy: p = 0.97, confidence: p = 1.3 × 10⁻⁶⁰), and (d) the version involving superimposed stimuli presented at different contrast levels (Fig. 3c; two-sided paired t-tests, accuracy: p = 0.82, confidence: p = 1.3 × 10⁻⁶²). Confidence for correct and incorrect trials is shown in Supplementary Fig. S3. e The latent ideal observer also accounted for the dissociation between type-1 and type-2 sensitivity. Results in (c) and (d) reflect probability density over 100 ideal observers (each based on distributions extracted by a separate trained VAE), with mean accuracy/confidence in each condition represented by circular markers, and maxima/minima represented by the upper/lower line; Results in (e) reflect mean d'/meta-d' over 100 ideal observers ± the standard of the mean; dotted black lines represent chance performance; ns indicates p > 0.05, ****p < 0.0001. Source data are provided as a Source Data file.